# What is the equation of the line tangent to  f(x)=(x-2)/x  at  x=-3 ?

May 11, 2018

$y = \frac{2}{9} x + \frac{7}{3}$

#### Explanation:

$f \left(x\right) = \frac{x - 2}{x}$ , $A = \mathbb{R}$*$= \left(- \infty , 0\right) \cup \left(0 , + \infty\right)$

$f ' \left(x\right) = \frac{\left(x - 2\right) ' x - \left(x - 2\right) \left(x\right) '}{x} ^ 2 = \frac{x - \left(x - 2\right)}{x} ^ 2 =$

$= \frac{x - x + 2}{x} ^ 2 = \frac{2}{x} ^ 2$

$f \left(- 3\right) = \frac{5}{3}$ , $f ' \left(- 3\right) = \frac{2}{9}$

$y - f \left(- 3\right) = f ' \left(- 3\right) \left(x + 3\right)$ $\iff$

$y - \frac{5}{3} = \frac{2}{9} \left(x + 3\right)$ $\iff$

$y = \frac{2}{9} x + \frac{7}{3}$